Problem: What is the remainder when $1 + 2 + 3 + 4 + \dots + 9 + 10$ is  divided by 8?
Explanation: Notice that we can pair many of these terms: \[1+7=2+6=3+5=8,\]so the remainder we want is the same as the remainder when $4+9+10$ is divided by 8.  We also see that this is the remainder when  \[4+1+2=7\]is divided by 8, so the answer is $\boxed{7}$.